Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount...

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Published in:Computer methods in applied mechanics and engineering Vol. 346; pp. 332 - 358
Main Authors: Jin, Bangti, Lazarov, Raytcho, Zhou, Zhi
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.04.2019
Elsevier BV
Subjects:
Computer simulation
Convolution
Error estimates
Evolution
Finite element method
Formulations
Galerkin method
Inverse problems
Mathematical analysis
Mathematical models
Nonsmooth solution
Numerical analysis
Numerical methods
Optimal control
Space–time formulation
Time-fractional evolution
Time-stepping
Finite element method
Time-stepping
Nonsmooth solution
Space–time formulation
Error estimates
Time-fractional evolution
ISSN:0045-7825, 1879-2138
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Abstract Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following topics of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space–time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.
AbstractList Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following topics of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space–time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.
Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α ∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following topics of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space–time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.
Author Jin, Bangti
Zhou, Zhi
Lazarov, Raytcho
Author_xml – sequence: 1
  givenname: Bangti
  orcidid: 0000-0002-3775-9155
  surname: Jin
  fullname: Jin, Bangti
  email: b.jin@ucl.ac.uk
  organization: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
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  givenname: Raytcho
  surname: Lazarov
  fullname: Lazarov, Raytcho
  email: lazarov@math.tamu.edu
  organization: Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
– sequence: 3
  givenname: Zhi
  surname: Zhou
  fullname: Zhou, Zhi
  email: zhizhou@polyu.edu.hk
  organization: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
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Keywords Finite element method
Time-stepping
Nonsmooth solution
Space–time formulation
Error estimates
Time-fractional evolution
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Snippet Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time,...
Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α ∈(0,1) in time,...
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SubjectTerms Computer simulation
Convolution
Error estimates
Evolution
Finite element method
Formulations
Galerkin method
Inverse problems
Mathematical analysis
Mathematical models
Nonsmooth solution
Numerical analysis
Numerical methods
Optimal control
Space–time formulation
Time-fractional evolution
Time-stepping
Title Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview
URI https://dx.doi.org/10.1016/j.cma.2018.12.011
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Volume 346
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